Crystallization is one of the most important separation and purification techniques employed industrially to produce a wide variety of materials. Crystallization is a common unit operation, used for producing a high purity solid phase from a fluid phase with a different composition. A key index of product quality is the product crystal size distribution (CSD). Specifically, for efficient downstream operations and product effectiveness, controlling the crystal size distribution can be critically important. Although there is wide awareness of the importance of crystallization, the optimal design and operation of crystallization processes still pose many problems such as the difficulty of developing a controlled cooling scheme or a strategy for antisolvent addition, which might lead to an undesired supersaturation. In short, there is a relative lack of systematic design procedures and predictive models to help overcome or avoid these problems.
The most well established approach to modeling of crystallizers is the population balance approach. See A. D. Randolph, and M. A. Larson, Theory of Particulate Processes (1988). Population balance equations (PBEs) provide a mathematical framework for dealing with processes involving formation of entities, growth, breakage or aggregation of particles, as well as dispersion of one phase into another phase. See E. J. Wynn and M. J. Hounslow, Integral Population Balance Equations for Growth, Chem. Eng. Sci. 52, 733 (1997); D. Ramkrishna and A. W. Mahoney, Population Balance Modeling. Promise for the Future, Chem. Eng. Sci. 57, 595 (2002). This mathematical approach follows the number of entities, such as solid particles, in such a way that their presence or occurrence predicts the behavior of the system under consideration. Population balance equations, however, are hyperbolic partial differential equations coupled with other ordinary differential and algebraic equations, and therefore they become large integro-partial differential algebraic equation (IPDAE) systems. These IPDAE systems usually cannot be solved analytically and therefore must be solved numerically. Various numerical techniques have been developed for solving IPDAE systems. See M. Wulkow, A. Gerstlauer, and U. Nieken, Modeling and Simulation of Crystallization Processes Using Parsival, Chem. Eng. Sci. 56, 2575 (2001). Each of these techniques has its own advantages and disadvantages usually requiring trade-offs between computational effort and accuracy of the model predictions, and requiring substantial implementation effort, specialized user training, and extensive user input.
Therefore, there is a need for a simpler modeling approach that addresses the disadvantages of prior models of crystallization processes.